[410] In connexion with his doctrine of the quantification of the predicate, Hamilton distinguishes between the figured syllogism and the unfigured syllogism. In the figured syllogism, the distinction between subject and predicate is retained, as in the text. By a rigid quantification of the predicate, however, the distinction between subject and predicate may be dispensed with; and such being the case there is no ground left for distinction of figure (which depends upon the position of the middle term as subject or predicate in the premisses). This gives what Hamilton calls the unfigured syllogism. For example:—Any bashfulness and any praiseworthy are not equivalent, All modesty and some praiseworthy are equivalent, therefore, Any bashfulness and any modesty are not equivalent; All whales and some mammals are equal, All whales and some water animals are equal, therefore, Some mammals and some water animals are equal. A distinct canon for the unfigured syllogism is given by Hamilton as follows:—“In as far as two notions either both agree, or one agreeing the other does not, with a common third notion; in so far these notions do or do not agree with each other.”
(1) UUU in figure 1 is valid:—
| All M is all P, | |
| All S is all M, | |
| therefore, | All S is all P. |
It will be observed that whenever one of the premisses is U, the conclusion may be obtained by substituting S or P (as the case may be) for M in the other premiss. 379
Without the use of quantified predicates, the above reasoning may be expressed by means of the two following syllogisms:
| All M is P, | All M is S, | ||
| All S is M, | All P is M, | ||
| therefore, | All S is P ; | therefore, | All P is S. |
(2) IUη in figure 1 is invalid, if some is used in its ordinary logical sense. The premisses are Some M is some P and All S is all M. We may, therefore, obtain the legitimate conclusion by substituting S for M in the major premiss. This yields Some S is some P.
If, however, some is here used in the sense of some only, No S is some P follows from Some S is some P, and the original syllogism is valid, although a negative conclusion is obtained from two affirmative premisses.
This syllogism is given as valid by Thomson (Laws of Thought, § 103); but apparently only through a misprint for IEη. In his scheme of valid syllogisms (thirty-six in each figure), Thomson seems consistently to interpret some in its ordinary logical sense. Using the word in the sense of some only, several other syllogisms would be valid that he does not give as such.[411]